Question

Arătați că x este un număr natural pătrat perfect:
a) x=(1+2+3+...+98)+49
b) x=1+3+5+...+225
c) x=1+3+5+...+2015
d) x=2+4+6+...+2016+1009

Answer 1

a) x=(1+2+3+...+98)+49 =98*99/2+49=49*99+49=49(99+1)=49*100=(7*10)^2

b) x=1+3+5+...+225=
adunam si scadem 2+4+....+224
=1+2+3+....225-(2+4+...224)=
=225*226/2-2(1+2+3+...+112)=
=225*113-2*112*113/2=
=225*113-112*113=
=113*(225-112)=
=113*113=
=113^2
 c) x=1+3+5+...+2015
adunam si scadem 2+4+...+2014
=1+2+3+....+2015-(2+4+....2014)=
=2015*2016/2-2(1+2+...+1007)=
=2015*1008-2*1007*1008/2=
=2015*1008-1007*1008=
=1008(2015-1007)=
=1008*1008=
=1008^2


d) x=2+4+6+...+2016+1009 =
=2(1+2+3+...+1008)+1009=2*1008*1009/2+1009=1008*1009+1009=1009(1008+1)=
=1009*1009=1009^2

^ inseamna la puterea